Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

4.2 Principle of Virtual Work 95

Bending Moment

The bending moment,M, acting on the member section in Fig. 4.5 produces a distribution of direct
stress,σ, through the depth of the member cross section. The normal force on the element,δA, cor-
respondingtothisstressisthereforeσδA.Againweshallsupposethatthestructureisgivenasmall
arbitraryvirtualdisplacementwhichproducesavirtualdirectstrain,εv,intheelementδA×δx.Thus,
thevirtualworkdonebythenormalforceactingontheelementδAisσδAεvδx.Hence,integrating
overthecompletecrosssectionofthemember,weobtaintheinternalvirtualwork,δwi,M,donebythe
bendingmoment,M,ontheelementallengthofmember:

δwi,M=

∫

A

σdAεvδx (4.18)

Thevirtualstrain,εv,intheelementδA×δxis,fromEq.(15.2),givenby

εv=

y

Rv

whereRvis the radius of curvature of the member produced by the virtual displacement. Thus,
substitutingforεvinEq.(4.18),weobtain

δwi,M=

∫

A

σ

y

Rv

dAδx

or,sinceσyδAisthemomentofthenormalforceontheelement,δA,aboutthezaxis,

δwi,M=

M

Rv

δx

Therefore,foramemberoflengthL,theinternalvirtualworkdonebyanactualbendingmoment,MA,
isgivenby

wi,M=

∫

L

MA

Rv

dx (4.19)

In the derivation of Eq. (4.19), no specific stress–strain relationship has been assumed, so that it is
applicabletoanonlinearsystem.Fortheparticularcaseofalinearlyelasticsystem,thevirtualcurvature
1/Rvmaybeexpressedintermsofanequivalentvirtualbendingmoment,Mv,usingtherelationshipof
Eq.(15.8):

1

Rv

=

Mv

EI

Substitutingfor1/RvinEq.(4.19),wehave

wi,M=

∫

L

MAMv

EI

dx (4.20)